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Beelzedad
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- The integral is not iterated integral. Will this fact prevent us from swapping the order of surface and volume integrals? Why? Why not?
Failure of Fubini's theorem for non-integrable functions[edit]
Fubini's theorem tells us that (for measurable functions on a product of σ-finite measure spaces) if the integral of the absolute value is finite, then the order of integration does not matter; if we integrate first with respect to x and then with respect to y, we get the same result as if we integrate first with respect to y and then with respect to x. The assumption that the integral of the absolute value is finite is "Lebesgue integrability", and without it the two repeated integrals can have different values.
A simple example to show that the repeated integrals can be different in general is to take the two measure spaces to be the positive integers, and to take the function f(x,y) to be 1 if x=y, −1 if x=y+1, and 0 otherwise. Then the two repeated integrals have different values 0 and 1.
Another example is as follows for the function
The iterated integrals
and
have different values. The corresponding double integral does not converge absolutely (in other words the integral of the absolute value is not finite):
One like this: ##\int_V f(x, y, z)dV##fresh_42 said:What do you mean by "not iterated integral"?
I don't see a difference. What is a "not iterated integral"? Is it simple versus multiple? ##\int_V f\,dV## and ##\int_{V_z}\int_{V_y}\int_{V_x} f\, dx\, dy\, dz ## is the same thing.Mark44 said:One like this: ##\int_V f(x, y, z)dV##
This can also be written as ##\iiint_V f(x, y, z)dV##. I've seen both styles in textbooks.
Here is an example of an iterated integral:
##\int_{y=0}^\pi\int_{x = 0}^\pi \sin(x + y)dx~dy##
One that is not an iterated integral...fresh_42 said:What is a "not iterated integral"?
Maybe, or maybe not. A double integral such as ##\int_R f\,dA## (where R is the region in the plane over which integration is to be performed) could be rewritten as two different iterated integrals: one in Cartesian form or one in polar form.fresh_42 said:Is it simple versus multiple? ##\int_V f\,dV## and ##\int_{V_z}\int_{V_y}\int_{V_x} f\, dx\, dy\, dz ## is the same thing.
I disagree. No calculus textbook that I've ever seen would write an integral like this: ##\int dx\,dy\,dz##.fresh_42 said:I would call this multiple or nested. It's not really an iteration. So the answer to my question is: A not iterated integral is a single integral. That means ##\int dV## versus ##\int dx\,dy\,dz## is more a linguistic issue than a mathematical.
Yes, I forgot to triple the integration symbol.Mark44 said:I disagree. No calculus textbook that I've ever seen would write an integral like this: ##\int dx\,dy\,dz##.
Yes, but how is this not purely notational? To distinguish it linguistically appears hair splitting to me.Again, it's the difference between this triple integral ##\int_D f(x, y, z) dV## or ##\iiint_D f(x, y, z) dV## (not iterated) and this iterated integral ##\int_{z = z_1}^{z_2}\int_{y = y_1}^{y_2}\int_{x = x_1}^{x_2} f(x, y, z) dx~dy~dz##.
Because the iterated form contains information about the order in which integration is to be performed, in addition to possibly distinguishing between Cartesian coordinates or polar/cylindrical coordinates are to be used.fresh_42 said:Yes, but how is this not purely notational?
The purpose of swapping the order of surface and volume integrals is to simplify the calculation process and make it more efficient. By rearranging the order, it may be possible to integrate over a smaller region or use simpler integrands, resulting in a faster and more accurate solution.
Swapping the order of surface and volume integrals is necessary when the original order of integration is not feasible or too complex. This can happen when the region of integration is difficult to define or when the integrand is too complicated to integrate in its original form.
The first step is to identify the limits of integration for the original order. Then, the region of integration is broken down into smaller, simpler regions. Next, the order of integration is swapped, and the new limits of integration are determined. Finally, the integrals are evaluated in the new order to obtain the solution.
Yes, there are limitations to swapping the order of surface and volume integrals. This method may not always be possible or may not result in a simpler solution. It is important to carefully analyze the integrand and the region of integration before deciding to swap the order.
Swapping the order of surface and volume integrals does not affect the final result. As long as the limits of integration are correctly determined and the integrals are evaluated correctly, the solution will be the same regardless of the order of integration.